Modal interactions in the dynamic response of isotropic and composite plates
Hadian, Mohammad Jafar
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Hamilton's principle and a third-order shear-deformation theory are used to derive a set of five coupled partial-differential equations governing the nonlinear response of composite plates. The reduction of these equations by using classical plate theory is discussed and the corresponding partial-differential equations governing both rectangular and circular plates are derived. Generalized Levy-type solutions are obtained for the problem of linear free vibrations and linear stability of shear-deformable cross-ply laminated plates. The governing equations are transformed into a set of first-order linear ordinary-differential equations with constant coefficients. The general solution of these equations is obtained by using the state-space concept. Then, the application of the boundary conditions yields equations for the natural frequencies and critical loads. However, a straightforward application of the state-space concept yields numerically ill-conditioned problems as the plate thickness is reduced. Various methods for overcoming this problem are discussed. An initial-value method with orthonormalization is selected. It is shown that this method not only yields results that are in excellent agreement with the results in the literature, but it also converges fast and gives all the frequencies and buckling loads regardless of the plate thickness. Further It is shown that the application of classical plate theory to thick plates yields inaccurate results. The influence of modal interactions on the response of harmonically excited plates is investigated in detail. The case of a two-to-one autoparametric resonance in shear-deformable composite laminated plates is considered. Four first-order ordinary-differential equations describing the modulation of the amplitudes and phases of the internally resonant modes are derived using the averaged Lagrangian when the higher mode is excited by a primary resonance. The fixed-point solutions are determined using a homotopy algorithm and their stability is analyzed. It is shown that besides the single-mode solution, two-mode solutions exist for a certain range of parameters. It is further shown that in the multi-mode case the lower mode, which is indirectly excited through the internal resonance may dominate the response. For a certain range of parameters, the fixed points lose stability via a Hopf bifurcation, thereby giving rise to limit cycle solutions. It is shown that these limit-cycles undergo a series of period-doubling bifurcations, culminating in chaos. Finally, the case of a combination resonance involving the first three modes of axisymmetric circular plates is studied. The method of multiple scales is used to determine a set of ordinary-differential equations governing the modulation of phases of the modes involved and that the excited mode is not necessarily the dominant one. Furthermore, it is shown that for a choice of parameters the multi-mode response loses stability through a Hopf bifurcation, resulting in periodically or chaotically modulated motions of the plate.
- Doctoral Dissertations